U.S. patent number 9,292,634 [Application Number 14/269,406] was granted by the patent office on 2016-03-22 for compensated molds for manufacturing ion exchange strengthened, 3d glass covers.
This patent grant is currently assigned to CORNING INCORPORATED. The grantee listed for this patent is CORNING INCORPORATED. Invention is credited to Izhar Zahoor Ahmed, Keith Raymond Gaylo, Jacob Immerman, John Richard Ridge, John Robert Saltzer, Jr., Ljerka Ukrainczyk.
United States Patent |
9,292,634 |
Ahmed , et al. |
March 22, 2016 |
Compensated molds for manufacturing ion exchange strengthened, 3D
glass covers
Abstract
Methods for compensating for the warp exhibited by
three-dimensional glass covers as a result of ion exchange
strengthening are provided. The methods use a computer-implemented
model to predict/estimate changes to a target three-dimensional
shape for the 3D glass cover as a result of ion exchange
strengthening. The model includes the effects of ion exchange
through the edge of the 3D glass cover. In an embodiment, the
inverse of the predicted/estimated changes is used to produce a
compensated (corrected) mold which produces as-molded parts which
when subjected to ion exchange strengthening have shapes closer to
the target shape than they would have had if the mold had not been
compensated (corrected).
Inventors: |
Ahmed; Izhar Zahoor (Painted
Post, NY), Gaylo; Keith Raymond (Painted Post, NY),
Immerman; Jacob (Elmira, NY), Ridge; John Richard
(Hammondsport, NY), Saltzer, Jr.; John Robert (Beaver Dams,
NY), Ukrainczyk; Ljerka (Painted Post, NY) |
Applicant: |
Name |
City |
State |
Country |
Type |
CORNING INCORPORATED |
Corning |
NY |
US |
|
|
Assignee: |
CORNING INCORPORATED (Corning,
NY)
|
Family
ID: |
51134241 |
Appl.
No.: |
14/269,406 |
Filed: |
May 5, 2014 |
Prior Publication Data
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Document
Identifier |
Publication Date |
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US 20140331716 A1 |
Nov 13, 2014 |
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Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
Issue Date |
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61820318 |
May 7, 2013 |
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Current U.S.
Class: |
1/1 |
Current CPC
Class: |
C03C
21/002 (20130101); H05K 5/03 (20130101); G06F
17/10 (20130101); C03B 23/0357 (20130101); G06F
30/20 (20200101) |
Current International
Class: |
C03B
23/03 (20060101); G06F 17/50 (20060101); C03C
21/00 (20060101); H05K 5/03 (20060101); C03B
23/035 (20060101) |
Field of
Search: |
;65/30.14,104,105,107 |
References Cited
[Referenced By]
U.S. Patent Documents
Foreign Patent Documents
Other References
Patent Cooperation Treaty International Notification of Transmittal
of the International Search Report and the Written Opinion of the
International Searching Authority, international application No.
PCT/US2014/036875: mailing date May 6, 2014, 10 pages. cited by
applicant.
|
Primary Examiner: Halpern; Mark
Attorney, Agent or Firm: Haran; John T.
Parent Case Text
CROSS-REFERENCE TO RELATED APPLICATIONS
This application claims the benefit of priority under 35 U.S.C.
.sctn.119 of U.S. Provisional Application Ser. No. 61/820,318 filed
on May 7, 2013 the content of which is relied upon and incorporated
herein by reference in its entirety.
Claims
What is claimed is:
1. A method of making a glass cover, said glass cover having a
target three-dimensional shape which comprises a planar central
portion and a perimeter portion which (i) borders at least part of
the planar central portion and (ii) extends out of the plane of the
planar central portion to provide the glass cover with three
dimensionality, said perimeter portion having a perimeter edge,
said method comprising: (I) providing a mold for forming the glass
cover, said mold having a molding surface; (II) producing the glass
cover using the mold of step (I); and (III) ion exchange
strengthening the glass cover produced in step (II); wherein the
method is characterized by: (a) the molding surface of the mold of
step (I) is produced based at least in part on computer-implemented
modeling which predicts/estimates changes to the target
three-dimensional shape that will result when the ion exchange
strengthening of step (III) is performed, and (b) said
computer-implemented modeling includes modeling the effects of ion
exchange through the perimeter edge.
2. The method of claim 1, wherein in said computer-implemented
modeling, the effects of ion exchange through the perimeter edge
are included in the modeling through a boundary condition at the
perimeter edge which permits ion permeation through the edge.
3. The method of claim 2, wherein in said computer-implemented
modeling, the boundary condition at the perimeter edge specifies a
constant ion concentration at the edge.
4. The method of claim 2, wherein in said computer-implemented
modeling: (i) ion diffusion is treated as thermal diffusion; and
(ii) based on that treatment in the modeling, the boundary
condition at the perimeter edge is treated as permitting heat flow
through the edge.
5. The method of claim 4, wherein in said computer-implemented
modeling, the boundary condition at the perimeter edge specifies a
constant temperature at the edge.
6. The method of claim 1, wherein the molding surface is produced
based at least in part on the inverse of the changes to the target
three-dimensional shape predicted/estimated by the
computer-implemented modeling.
7. The method of claim 1, wherein in said computer-implemented
modeling a mesh is employed and the mesh has a size at the
perimeter edge in the range of five to ten microns.
8. The method of claim 1, wherein in said computer-implemented
modeling, ion exchange through the perimeter edge results in
predicted/estimated changes to the target three-dimensional shape
of a larger magnitude than ion exchange through the remainder of
the outer surface of the glass cover.
9. The method of claim 1, wherein the glass cover produced in step
(II) is for a portable electronic device.
Description
FIELD
This disclosure relates to three-dimensional glass covers (3D glass
covers) for electronic devices, such as, mobile or handheld
electronic devices. More particularly, the disclosure relates to
three-dimensional glass covers that have been ion exchange
strengthened (IOX strengthened) and to the molds used to make such
glass covers.
BACKGROUND
FIG. 1 shows a representative, non-limiting, shape for a 3D glass
cover (also known in the art as a "3D cover glass"), which can be
used with an electronic device, such as, a telephone, television,
tablet, monitor, or the like. As shown in this figure, 3D glass
cover 100 includes: (i) a planar central portion 101, (ii) a
perimeter portion 102, and (iii) a perimeter edge 103.
Planar central portion 101 is flat or nearly flat, i.e., its radius
of curvature is at least 300 millimeters. Perimeter portion 102
extends out of the plane of the planar central portion 101, thus
providing the glass cover with an overall three-dimensional shape,
as opposed to a two-dimensional shape. Although as shown in FIG. 1,
perimeter portion 102 completely surrounds central portion 101, in
some embodiments, the perimeter portion can extend around only a
portion of the central portion, e.g., for a glass cover having a
rectangular shape, less than all four sides of the glass cover can
include a perimeter portion, e.g., two sides can have a perimeter
portion and the other two sides can be flat or substantially flat.
Likewise, to be three-dimensional, a glass cover in the form of a
disc or saucer only needs to have a portion of its flat or nearly
flat central portion transition into a perimeter portion which
extends out of the plane of the flat or nearly flat central
portion.
As will be evident, the shape of a 3D glass cover can vary widely
depending on the desires of the designer of the device with which
the 3D glass cover will be used. Thus, the 3D glass cover can have
a variety of overall shapes and can include central portions and
perimeter portions of various sizes and shapes, and can employ
transitions of various configurations between the central and
perimeter portions. Commonly-assigned U.S. application Ser. No.
13/774,238 entitled "Cover Glass Article" filed Feb. 22, 2013,
published as U.S. Patent Application Publication No. 2013/0323444,
the contents of which are incorporated herein by reference,
provides various representative dimensions for 3D glass covers, as
well as descriptions of typical applications for the covers. The
molding technology disclosed herein can be used with 3D glass
covers of these types, as well as other types now known or
subsequently developed.
The transverse dimension (thickness) of perimeter edge 103
corresponds to the thickness of the glass from which the glass
cover is made, which is typically less than 1 millimeter, e.g., 0.8
millimeters or less. Because of this small thickness, prior to the
present disclosure, it had been believed that stress changes at the
edge could be ignored in predicting changes in the overall shape of
a 3D glass cover as a result of ion exchange (IOX) strengthening.
In particular, on an area basis, the perimeter edge of a typical
glass cover amounts to less than about two percent of the overall
area of the part. Hence, the number of ions exchanged through the
edge is only a small fraction of the total number of ions
exchanged, thus making it reasonable to assume that relative to the
total number of ions exchanged, those few ions would have little
effect on the structural behavior of the part.
In fact, in accordance with the present disclosure, it has been
surprisingly found that ion exchange at the edge not only has a
substantial effect on the overall shape, but its effect is in many
cases greater than the effect of ion exchange on the rest of the
part. Thus, although only a small number of ions move through the
edge relative to the total number of ions that move through the
surfaces of the rest of the part, those edge-traversing ions are
critical to the shape changes exhibited by 3D glass covers as a
result of IOX strengthening. In terms of commercial value, this
discovery permits manufacturers of 3D glass covers to effectively
and efficiently meet tolerance requirements of customers for those
covers. In particular, as detailed below, it allows manufacturers
of 3D glass covers to produce molds for making those covers which
accurately compensate for the changes in shape which the cover will
exhibit when IOX strengthened. The technology thus, represents a
valuable contribution to the ability of designers to create
aesthetically pleasing designs for 3D glass covers and the
manufacturers of the covers to accurately produce the shapes
envisioned by the designers.
SUMMARY
In accordance with a first aspect, a method is disclosed of making
a glass cover (100), said glass cover (100) having a target
three-dimensional shape which comprises a planar central portion
(101) and a perimeter portion (102) which (i) borders at least part
of the planar central portion (101) and (ii) extends out of the
plane of the planar central portion (101) to provide the glass
cover (100) with three dimensionality, said perimeter portion (102)
having a perimeter edge (103), said method comprising:
(I) providing a mold (200) for forming the glass cover (100), said
mold (200) having a molding surface (208);
(II) producing the glass cover (100) using the mold (200) of step
(I); and
(III) ion exchange strengthening the glass cover (100) produced in
step (II);
wherein the shape of the molding surface (208) of the mold (200) of
step (I) is based at least in part on a computer-implemented model
which predicts/estimates changes to the target three-dimensional
shape as a result of the ion exchange strengthening of step (III),
said computer-implemented model including the effects of ion
exchange through the perimeter edge (103).
Optionally, the glass cover can be annealed to relieve residual
thermal stress between steps (II) and (III).
In accordance with a second aspect, a computer-implemented method
is disclosed for predicting/estimating changes in the shape of a
three-dimensional glass cover (100) after ion exchange
strengthening, said glass cover (100) comprising a planar central
portion (101) and a perimeter portion (102) which (i) borders at
least part of the planar central portion (101) and (ii) extends out
of the plane of the planar central portion (101) to provide the
glass cover (100) with three dimensionality, said perimeter portion
(102) having a perimeter edge (103), said method comprising
employing a boundary condition at the perimeter edge (103) which
permits ion permeation through the edge so as to model the effects
of ion exchange through the perimeter edge (103) on the shape of
the glass cover (100).
In embodiments of the above methods, ion diffusion is treated as
thermal diffusion and the boundary condition at the perimeter edge
permits heat flow through the edge.
The reference numbers used in the above summaries of the aspects of
the invention are only for the convenience of the reader and are
not intended to and should not be interpreted as limiting the scope
of the invention. More generally, it is to be understood that both
the foregoing general description and the following detailed
description are merely exemplary of the invention and are intended
to provide an overview or framework for understanding the nature
and character of the invention.
Additional features and advantages of the invention are set forth
in the detailed description which follows, and in part will be
readily apparent to those skilled in the art from that description
or recognized by practicing the invention as exemplified by the
description herein. The accompanying drawings are included to
provide a further understanding of the invention, and are
incorporated in and constitute a part of this specification. It is
to be understood that the various features of the invention
disclosed in this specification and in the drawings can be used in
any and all combinations.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a perspective view of a representative 3D glass
cover.
FIG. 2 is a schematic, cross-sectional diagram illustrating a
representative mold for producing a 3D glass cover.
FIG. 3 shows a cross-sectional slice of an example saucer-shaped
part. The "edge" of the part is indicated by arrows.
FIG. 4 shows a cross-sectional slice of a bent shape for the part
of FIG. 3. The x-axis and y-axis numbers are arbitrary in this
figure.
FIG. 5 is a schematic diagram showing an original (solid lines) and
a warped (dashed lines) axisymmetric shape.
FIG. 6 is a schematic diagram illustrating a geometry for which an
analytic solution for ion diffusion can be obtained.
FIG. 7 is a schematic diagram showing finite element analysis
results for the geometry shown in FIGS. 3 and 4. The straight lines
show where the glass was before ion exchange.
FIG. 8 is a schematic diagram showing finite element analysis
results for the geometry shown in FIGS. 3 and 4. The straight lines
show where the glass was before ion exchange.
FIG. 9 is a schematic diagram showing finite element analysis
results for the geometry shown in FIGS. 3 and 4. In particular,
this figure magnifies the cover's edge to show that only this part
of the cover was ion exchanged. The straight lines show where the
glass was before ion exchange.
FIG. 10 is a schematic, cross-sectional view of a saucer-shaped
part identifying various geometric variables.
FIG. 11 is a schematic diagram showing finite element analysis
results for six cases, where .alpha.=10.degree. for the left-hand
column and .alpha.=90.degree. for the right-hand column. The
straight lines show where the glass was before ion exchange.
FIG. 12 is a graph showing warp of the saucer part of FIG. 10 vs.
angle .alpha.. At .alpha.=90.degree. the height h of FIG. 10 is 7.7
mm.
FIG. 13 is a graph showing warp of the saucer part of FIG. 10 vs.
angle .beta..
FIG. 14 is graph showing warp of the saucer part of FIG. 10 vs. the
length x of the out-of-plane, perimeter portion of the saucer.
FIG. 15 is graph showing warp of the saucer part of FIG. 10 vs. the
length of the flat central portion of the saucer.
FIG. 16 is a schematic, perspective view of a portion of a
representative 3D glass cover illustrating representative meshing
over the major surfaces of the glass cover suitable for use in
predicting the warp of the part as a result of IOX
strengthening.
FIG. 17 is a schematic, perspective view of a portion of a
representative 3D glass cover illustrating representative meshing
over the edge of a 3D glass cover whose IOX-warp behavior is to be
predicted/estimated.
FIG. 18 is a flowchart illustrating a mold contour correction
method in accordance with an embodiment of the disclosure.
FIG. 19 is a schematic diagram illustrating the importance of
including the edge of a 3D glass cover when modeling warp of the
cover during IOX strengthening.
FIG. 20 is a graph illustrating IOX warp with respect to CAD shape
(target shape) in the curved region (perimeter portion) of a 3D
glass cover.
FIG. 21 is a graph illustrating IOX warp with respect to CAD shape
(target shape) in the flat region (central portion) of a 3D glass
cover.
FIG. 22 is a graph of measured data for a 3D glass cover as-molded
(curve 108) and post-IOX (curve 109) where the mold was
uncorrected.
FIG. 23 is a graph of measured data for a 3D glass cover as-molded
(curve 110) and post-IOX (curve 111) using the same mold as FIG. 22
but after correction in accordance with the present disclosure.
The warp shown in FIGS. 4-5, 7-9, 11, and 19 is not drawn to scale
but has a magnified y-axis scale for purposes of illustration.
DETAILED DESCRIPTION
As discussed above, the present disclosure is concerned with
methods of producing IOX-strengthened, 3D glass covers having a
shape that corresponds closely to that specified by the designer of
the cover (the target shape; also known as the "CAD shape" in cases
where the shape is specified through a CAD drawing). The disclosure
also relates to methods for designing molds having mold surfaces
which compensate for (correct for) the changes in shape which a 3D
glass cover undergoes when IOX strengthened.
In particular, after ion exchange, a 3D glass cover warps in the
range of 10-150 microns depending upon the part shape. This happens
because of dilatation in IOX: smaller ions, e.g., sodium ions, are
replaced by larger ions, e.g., potassium ions, in the outer
.about.40 to .about.100 microns of glass thickness. This causes the
glass's dimensions to increase. In flat glass, this dimensional
growth in IOX is about 0.04%. As will be discussed below, in glass
having a 3D shape, the behavior of the glass is much more
complicated than a simple dimensional growth.
Shape deviation as a result of IOX warp is not desirable as
customer specifications are typically .+-.100 microns. In order to
compensate for this IOX warp, in accordance with the present
disclosure, mold contour corrections are used which cause the
as-molded part to deviate from the target shape in a manner such
that after IOX strengthening, the shape of the part is closer to
the target shape than it would have been without the corrections.
That is, the as-molded part is moved away from the target shape so
that the IOX strengthened part will be closer to that shape.
Because IOX warp depends on the details of the overall shape of the
glass cover, as well as on details of the shape and thickness of
the cover's edge, in general, individualized mold correction values
are needed for each 3D glass cover that is to be produced. In
accordance with an aspect of the disclosure, these correction
values are obtained by transforming the IOX warp problem into a
thermal diffusion problem, thus allowing the IOX problem to be
solved using commercially-available software, e.g., ANSYS.RTM.
software sold by ANSYS Inc., 275 Technology Drive, Canonsburg, Pa.
15317, USA, which employs thoroughly-tested, state-of-art finite
element and graphical display techniques. Also, target shapes and,
in particular, CAD-formatted target shapes, can be readily inputted
to such commercially-available software. In practice, using the
technology disclosed herein, mold contour corrections can be
rapidly developed without the need for repeated iterative changes
to physical molds. Indeed, in many cases, a single iteration will
be sufficient to reduce deviation from CAD after IOX to
approximately .+-.10 microns, thus allowing the 3D glass covers to
meet customer specifications.
In some embodiments, the 3D glass cover is made from a 2D glass
sheet using a thermal reforming process such as described in U.S.
Patent Application Publications Nos. 2010/0000259 and 2012/0297828,
both incorporated herein by reference. In some embodiments, the 2D
glass sheet is made by a fusion process, although 2D glass sheets
made by other processes, such as by float or rolling processes, may
also be used.
FIG. 2 is a schematic, cross-sectional diagram illustrating a
representative mold suitable for use in a thermal reforming process
of the type disclosed in the above-referenced patent applications.
In this figure, mold 200 includes a mold body 202 having a top
surface 206 and a cavity 204. The cavity is open at the top surface
206 and its bottom comprises molding (shaping) surface 208. Molding
surface 208 has a surface profile which, in accordance with the
present disclosure, is corrected to compensate for IOX warping. As
can be appreciated, the profile of molding surface 208 will vary
from that shown in FIG. 2 depending on the specifics of the 3D
glass cover that is to be made.
As shown in FIG. 2, mold body 202 can include one or more slots
and/or holes 210 (hereinafter referred to as "apertures") extending
from the bottom surface 215 of the mold body to the molding
surface. Apertures 210 are arranged to provide communication
between the exterior of the mold and the molding surface. In one
example, the apertures are vacuum apertures. That is, the apertures
can be connected to a vacuum pump or other device (not shown) for
providing vacuum to the cavity 204 through the molding surface
208.
FIG. 2 also shows a flat glass plate 218 having a portion 220
located over cavity 204. Briefly, in forming a 3D glass cover using
a mold of the type shown in FIG. 2, heat is applied to plate 218 so
that it sags into cavity 204, while a vacuum is applied to conform
the softened glass to the shape that has been machined into molding
surface 208. To withstand the temperatures associated with this
process, mold 200 may be made of a heat-resistant material. As an
example, the mold may be made of high temperature steel or cast
iron. To extend the life of the mold, the molding surface may be
coated with a high-temperature material that reduces interaction
between the mold and the glass making up the glass cover, e.g., a
chromium coating.
After removal from the mold and such post-molding processing as is
desired, e.g., annealing, the molded 3D glass cover is subjected to
ion exchange strengthening. Various ion exchange techniques now
known or subsequently developed can be used depending on the
specific performance requirements of the 3D glass cover and the
composition of the glass making up the cover. Examples of such
processes can be found in U.S. Provisional Application No.
61/666,341 entitled "Methods For Chemically Strengthening Glass
Articles" filed Jun. 29, 2012, to which U.S. application Ser. No.
13/923,837 claims the benefit of, the contents of which are
incorporated herein by reference. Examples of glass compositions
suitable for ion exchange strengthening can be found in U.S. Pat.
Nos. 4,483,700, 5,674,790, 7,666,511, and 8,158,543; and U.S.
Patent Application Publications Nos. 2009-0142568, 2011-0045961,
2011-0201490, 2012-0135226, and 2013-0004758, the contents of which
are incorporated herein by reference.
In broad outline, ion exchange strengthening involves treating the
formed glass article by submersing it in a salt bath at an elevated
temperature for a predetermined period of time. The process causes
ions from the salt bath, e.g., potassium ions, to diffuse into the
glass while ions from the glass, e.g., sodium ions, diffuse out of
the glass. Because of their different ionic radii, this exchange of
ions between the glass and the salt bath results in the formation
of a compressive layer at the surface of the glass which enhances
the glass's mechanical properties, e.g., its surface hardness. The
effects of the ion exchange process are typically characterized in
terms of two parameters: (1) the depth of layer (DOL) produced by
the process and (2) the final maximum surface compressive stress
(CS). Values for these parameters are most conveniently determined
using optical measurements, and commercial equipment is available
for this purpose, e.g., instruments sold by Frontier Semiconductor
and Orihara Industrial Company, Ltd.
As discussed above, in accordance with the present disclosure, it
has surprisingly been found that the ion exchange that occurs at
the out-of-plane edge (or out-of-plane edges) of a 3D glass cover
is a major driver, indeed, in most cases, the major driver, of warp
of the cover resulting from the IOX process. Although not wishing
to be bound by any particular theory of operation, this effect can,
in retrospect, be interpreted in terms of bending moments. (Note
that for ease of reference, the following analysis uses the phrase
"bending moment" instead of the more precise phrase "bending moment
per unit length.")
One of the simplest examples of a bending moment occurs in a long
thin bar in which there is strain varying only in the thickness
direction, taken to be the z-direction. The bending moment integral
in this case is defined by:
.times..times..times..intg..times..function..times.d ##EQU00001##
where E is the Young's modulus of the glass beam, B is the "lattice
dilation coefficient" (the factor that converts concentration of
exchanged ions into strain), C(z) is the concentration of the
larger ion minus its value in the base glass, and the depth z
varies from -h/2 on the bottom surface to +h/2 on the top
surface.
Using this definition, the final shape of the beam after ion
exchanging the top and bottom surfaces to produce the concentration
profile C(z) is given by:
.function..times..times..times..times..times..times..intg..times..functio-
n..times..times.d.times. ##EQU00002##
This result, which ignores any rigid body motion, is a valid
description of the z-component of displacement (called w here)
along the centerline of the beam (that is, through the middle) as a
function of length along the beam where x=0 at the center of the
beam. The final expression shows that the bending of the beam is
independent of Young's modulus E and depends only on the lattice
dilation coefficient B, the beam height h, and the concentration
profile. A similar derivation for thermal diffusion can be found in
B. A. Boley and J. H. Weiner Theory of Thermal Stresses, Dover
Publications, 1988, p. 279 et seq. (hereinafter referred to as
"Boley/Weiner").
If the concentration profile is symmetric about the center z=0,
then the bending moment integral of Eq. (1) is zero and there is no
bending of the beam. If the concentration profile is asymmetric,
e.g. as in the case of float glass due to its asymmetry of
manufacture, then the bending moment integral will be nonzero and
the beam will take on a parabolic shape as given by Eq. (2). For
example, if more ions are exchanged in the top half than in the
bottom half, then the integral for M will be a positive number and
the bending will be in a negative sense according to the minus sign
in Eq. (2). This makes intuitive sense because where more ions
enter there is greater expansion of the glass which causes the beam
to bend away from this surface and toward the opposite surface.
In order to proceed to a 3D shape that bears some resemblance to
practical parts of interest we consider an axisymmetric 3D case of
a disk-shaped part that turns sharply at its perimeter. This is
meant to simplify realistic cases that typically have a rectangular
shape (see, for example, FIGS. 1 and 16) and a bend radius
.gtoreq.1 mm (see, for example, FIGS. 1, 16 and 17), whose
magnitude may be a further variable of IOX warping. As we show
below, the warping of real 3D parts as a result of ion exchange of
the glass edge is driven primarily by the same mechanism we are
about to describe; this mechanism involves expanding glass at the
edge pushing to a larger size by bending the underlying part.
Bending is driven by a nonzero bending moment as in the simple
example above.
The simplified geometry is shown in FIG. 3. Only a slice of the
object is shown, the full object being obtained by rotating this
slice by 180.degree. around an axis that extends vertically through
its center. When the edges of this part are ion exchanged, the
region of glass near the points of the arrows in FIG. 3 wants to
expand relative to nearby glass that is further from the surface.
If the circumference of the ring created by the edge could be made
larger it would relieve some of the elastic energy built up by the
free strain from ion exchange. For example, imagine the part being
pushed down onto the surface of a sphere so that it takes on a
shape shown in FIG. 4.
If the part could take on the shape of FIG. 4, then the
circumference along the edge of the glass actually becomes larger.
Because this would relieve some elastic energy, the part will tend
to bend in this fashion even though bending costs elastic energy.
The resistance to bending of the entire part can be smaller than
the energy gained by enlarging the circumference at the edge
depending on thickness and other details.
Quantitatively, the enlarging of the circumference is given as
follows. If the original radius of the part prior to bending is a,
then the original circumference of the edge is: C.sub.0=2.pi.a.
(3)
If the radius of curvature of the bent part is R (which is
typically much larger than a) then the new circumference of the
in-plane glass (not the glass located at the arrow tips in FIG. 3,
but the glass located at the right-angle turn) is, through second
order:
.times..pi..times..times..times..times..times..times..theta..times..times-
..pi..times..times..times..times..times..apprxeq..times..pi..times..times.-
.function..times..apprxeq. ##EQU00003##
This says the bending of the edge does very little to the original
circumference of glass that was in the plane of the flat part. The
glass at the very edge, however, gets the benefit of being out of
the plane of the flat part by a little bit. Suppose it sticks up by
an amount .delta.. The new circumference of the edge glass is
then:
.times..pi..times..times..delta..times..times..times..times..theta..times-
..times..pi..times..times..delta..times..times..times..times..pi..times..t-
imes..delta..times..times..pi..times..times..times..delta.
##EQU00004##
This says that bending of the part sticking out of the plane by
.delta. adds additional circumference that is proportional to
.delta.. This provides a simple mechanism for relieving stress by
taking advantage of slightly bending the part, but it is only
relevant when a portion of the glass is out of plane and undergoes
free strain from ion exchange. This also says that the edge region
alone, the region just addressed in this analysis, is responsible
for a driving force that tends to bend the disk-shaped part in
order to lower its elastic energy.
The driving force is related to elastic energy but can be expressed
in terms of forces to gain additional insight. FIG. 5 shows another
depiction of the original and warped shape. When the top edge is
ion exchanged, if the edge were not constrained by being connected
with the rest of the saucer-shaped part, the edge would expand
radially by roughly an amount: .DELTA.r=RB.DELTA.C (6) where
.DELTA.C is the change in exchanged ion concentration and B is the
lattice dilation coefficient mentioned in Eq. (1) above.
If unconstrained, the edge would expand freely in its own plane.
However, because the edge is constrained by the vertical (or upward
in general) part of the glass adjoining it, and it is further
constrained by the rest of the part to which the vertical part is
connected, there will be a horizontal force and a bending moment
generated. This provides a driving force to bend the part in order
to accommodate the free strain of Eq. (6).
The above example can be examined in greater mathematical detail
since an axisymmetric plate with a concentration that varies only
in the radial direction r and the thickness direction z with free
boundary conditions (and ignoring gravity) can be solved in certain
cases (cf. Boley/Weiner, p. 389 et. seq.). The differential
equation to be solved is (cf. Boley/Weiner, Eq. (12.2.16)):
.times..gradient..times..times..gradient..times..times..times..times..tim-
es..times. ##EQU00005##
In these equations, w(r) is the vertical displacement of the part
along its midplane as a function of radius, D is sometimes called
the bending or flexural rigidity, E is Young's modulus (which will
divide out so it does not enter the calculation), h is the original
height of the part, and v is the Poisson ratio. M is the bending
moment given by the integral
.function..times..times..times..intg..times..function..times..times.d
##EQU00006## where B is the lattice dilation coefficient converting
concentration to strain and C(r,z) is the concentration as a
function of radius and vertical location. This is the same as Eq.
(1) only with radial dependence.
Eq. (7) can be rewritten in the form
.times.dd.function..times.dd.times..times.dd.times..times.dd.times..times-
..times.dd.function..times.dd.times..times..intg..times..function..times..-
times.d ##EQU00007##
The boundary condition along the outer radius of the part (r=a) for
a free edge is given by (cf. Boley/Weiner, Eq. (12.4.26)):
d.times.d.times.dd.times.dd.times..times.dd.times..times.dd.times.dd
##EQU00008##
To make a simple and solvable example, we consider a uniformly
expanding region (representing an ion exchanged region, but
simplified to uniform expansion or constant concentration)
expressed by the concentration profile:
.function.<<.times..times..times..times..ltoreq..delta..times.
##EQU00009##
Notice that this introduces the same kind of asymmetry relative to
the midplane as in the case studied earlier in that the ion
exchanged region is only near the top surface, not symmetrically
near the bottom surface as well. This is what gives a nonzero
bending moment. A cross-sectional slice is sketched in FIG. 6. The
region with uniform expansion is shown as a darker portion at the
top outer edge. After some algebra the above equations are solved
to give the following expression for warp vs. radius:
.function..times..times..times..times..times..delta..function..delta..tim-
es. .times..times..times.<.times..times..function..gtoreq.
##EQU00010##
Over most of the radius of the part, that is, inside r.sub.1, the
warp can be simplified to the form:
.function..times..times..times..times..times..delta..function..delta..tim-
es..times..times. ##EQU00011##
This shows that the originally flat part takes on a parabolic shape
with a magnitude of warp that is proportional to .delta. (for small
.delta.) and proportional to the free strain BC and approximately
inversely proportional to the square of the height h (one factor
approximately divides out with the factor of (h-.delta.) in the
numerator). Going exactly to the edge, the warp is slightly altered
from the parabolic shape, but the overall scale is about the same.
From the sign of the warp, we see that positive strain BC>0
creates a negative sense of warp, that is, the part becomes concave
down as shown elsewhere, e.g., FIG. 4.
The overall warp expressed as a single number can be taken to be
the maximum vertical distance from the center to the new edge
location. In Eq. (13) this is just w(r=a), which gives the final
result:
.times..function..times..times..times..times..times..times..delta..functi-
on..delta..times. .times..times..times..times..times..function.
##EQU00012##
These analytic results provide insight into the roles of (1)
bending moment (Eq. (9)) and (2) asymmetry of concentration in
generating warp. To analyze more realistic cases, we switch from
analytic methods to numerical analysis. Specifically, we continue
to treat basically the same case but by using numerical finite
element analysis, specifically, the ANSYS.RTM. commercial finite
element software referred to above, we are free to study more
complex geometries and to use a realistic representation of the
concentration profile.
FIGS. 7-9 show the final state of warp as predicted/estimated by a
finite element analysis for the same disk-shaped object studied
above. The straight lines in these figures show where the glass was
before ion exchange. FIG. 9 shows the concentration profile
superimposed on the distorted geometry (see the uppermost part of
the distorted geometry). It is important to note that in this
calculation only the very edge was ion exchanged, yet due to its
impact on bending moments, this was enough to drive all the
distortion shown. This surprising result is crucially important for
capturing the bending and warping of a part resulting from ion
exchange.
With the importance of ion exchange on the edge of the part having
been established, we now turn to examples from finite element
analysis showing various trends in warp behavior with changes in
part geometry. FIG. 10 shows the variables under study.
FIG. 11 shows results for angle .alpha. of 10.degree. (left panels)
and 90.degree. (right panels) with 3 cases each: all surfaces ion
exchanged (top), all surfaces exchanged except the top edge
(middle), and only the top edge (bottom). On the left, with smaller
.alpha., it is very clear that the warping effect is dominated by
ion exchanging only the edges (bottom left panel). Practical cases
of interest are usually closer to this low angle bending case;
hence we focus on the edge exchange as a critical driver for
out-of-plane warp. In the case where the edge is exactly at a right
angle to the rest of the part, as shown in the right panels,
additional terms come into play so that the full vertical part and
not just its edge contribute to warping. Even in this case, ion
exchange of the edge is responsible for about a third of the
warp.
As .alpha. varies from very small angles with a nearly flat part to
90.degree. there is an interplay of effects so the total amount of
warp is not monotonic in .alpha.. Results are shown in FIG. 12. At
very small .alpha. we can explain the trend because at .alpha.=0
there is no longer any asymmetry and no driving force for warp. At
low .alpha., warp grows proportional to .alpha.. At somewhere near
10.degree. another effect arises: as .alpha. grows larger the
moment arm for the bending moment to bend the part becomes smaller
(the outer radius of the part is growing smaller as .alpha.
increases) so the trend reaches a maximum and turns around.
Additional subtleties could be examined but the main point is that
multiple considerations are at work to create a final part shape
and a carefully constructed and accurate model is critical to
getting useful results.
When .beta.=90.degree. we have the edge we have been discussing up
to this point; other values of .beta. introduce a bevel as shown in
FIG. 10. The trend of overall warp with .beta. is given in FIG. 13.
Tilting away from about 90.degree. in either sense increases the
warp. This is believed to be a result of the increased surface area
afforded by the bevel which allows more ions to exchange in the
sample, driving a larger strain or a larger overall BC as mentioned
in the analytic model.
The length of the out-of-plane feature has a non-monotonic impact
on warp similar to that of angle .alpha.. Results are shown in FIG.
14, where "curve length" represents the length of the perimeter
portion of the part. When the length of the out-of-plane curve
approaches 0 then once again there is no asymmetry of geometry and
no bending moment from ion exchange and no driving force for warp,
so the warp must go to zero as x approaches 0. At a length of
around 2 mm in this example the warp reaches a maximum value and
then starts to fall again. This is believed to be a result of
additional rigidity provided by a longer out-of-plane curve. At
some point with large enough x the out-of-plane curve can become
rigid enough to prevent the flat base from warping.
The effects of flat length are shown in FIG. 15. From Eq. (15) we
expect the overall warp to rise roughly with the square of the
radius a (or the square of the flat length in this case). As can be
seen in FIG. 15, the main trend is parabolic as expected from the
analytic model.
With the above understanding of the balance of forces, bending
moments, and ion exchange-induced stresses, we now turn to the
practical problem of predicting/estimating the warping of 3D glass
covers as a result of ion exchange strengthening. Having an
accurate model for this effect, we then subtract off the effect
from the original mold (e.g., an original mold which is identical
to the target shape), so that after ion exchange warping the final
shape agrees with the target shape. To manage 3D shapes using
finite element analysis, it is convenient to use well-established
computer programs. Commercial and open-source software packages are
usually designed to calculate thermal stresses and thermal warp and
not ion exchange stresses or ion exchange warp, so some additional
understanding is needed in order to create a thermal problem that
mimics the details of the ion exchange problem.
In accordance with the present disclosure, this is done using a
mathematical analogy. The mathematical analogy between
concentration/stress and temperature/stress exploits the fact that
both concentration and temperature obey the same diffusion
equation. In three dimensions, the governing equation for mass
diffusion is:
.differential..differential..function..differential..times..differential.-
.differential..times..differential..differential..times..differential.
##EQU00013## for the case of constant diffusivity D, where C
represents concentration of the diffusing species. The
three-dimensional boundary conditions for the cases of interest are
C(x,y,z,t=0)=C.sub.base (17) at all points (x,y,z) inside the
sample at the initial time and C(x,y,z,t)=C.sub.surf (18) on all
the surfaces at all times.
When we say "ion exchange at the edge must be included in the
calculation" we mean that the C.sub.surf boundary condition must be
applied at the thin edge that has been discussed previously and
which is shown in FIG. 9 for the saucer example. Moreover, a
sufficiently fine mesh needs to be used to model the edge. FIGS. 16
and 17 show examples of suitable meshes for use in modeling 3D
glass covers. As can be seen in these figures and, in particular,
in FIG. 17, the meshing for edge 103 is finer than that used for
the cover's central portion 101 and for the major parts of its
perimeter portion 102. A suitable mesh spacing for the edge is in
the range of 5-10 microns.
By Saint-Venant's Principle one would have argued that edge effects
should not have much impact far from the edges, and one would have
ignored ions entering this small region. Specifically, instead of
using the above boundary condition, one would simply define the
edge as impermeable to ions. However, as shown above, the edges
introduce an unbalanced bending moment that becomes primarily
responsible for the overall ion exchange-induced warping of the
part, and thus cannot be ignored.
In three dimensions, local "free strains" (before elastic
relaxation or before incorporating material continuity or
compatibility) are equal to BC for lattice dilation coefficient B.
(Recall B is the coefficient that converts concentration to
strain.) Three-dimensional stress calculations use these initial
strains along with stress boundary conditions and compatibility
conditions. The final 3D stresses and strains are calculated based
on these initial conditions and boundary conditions using the
techniques employed with thermally induced stresses and
strains.
In three dimensions, the governing equation for thermal transfer
(or equivalently, heat diffusion) is:
.differential..differential..rho..times..times..function..differential..t-
imes..differential..differential..times..differential..differential..times-
..differential..rho..times..times. ##EQU00014## where T is the
temperature, k is the thermal conductivity, .rho. is the density,
C.sub.y is the specific heat capacity at constant pressure, and Q
is a source of heat added per time per unit volume. This is the
equation that is solved by commercial finite element software such
as the ANSYS.RTM. software referred to above.
Typical boundary conditions could assign an initial temperature
that is uniform throughout the body and a surface temperature that
is fixed throughout the time evolution of the calculation. Free
strains are given by the coefficient of thermal expansion .alpha.
times the temperature, .alpha.T. Temperature plays the same role as
concentration and coefficient of thermal expansion .alpha. plays
the role of lattice dilation coefficient B. The three-dimensional
boundary conditions are handled by replacing concentration with
temperature.
To go further, Eq. (19) needs to be converted into Eq. (16). There
is nothing in the ion exchange problem analogous to a heat source
so in the thermal problem we set Q=0. Next we replace the quotient
k/(.rho.C.sub.p) by a single constant known as the thermal
diffusivity. We can set k, the thermal conductivity in the thermal
problem, exactly to D, the diffusivity in the mass transfer
problem, by setting .rho.C.sub.p=1. Given that Q=0, the actual
values of .rho. and C.sub.p are irrelevant but they must be chosen
so as to keep .rho.C.sub.p=1. The relationships between physical
constants of the mass transfer and thermal problems are summarized
in Table 1. The values of the physical properties listed in Table 1
can be readily determined for any particular glass by skilled
persons using measurement techniques known in the art.
The analogy is completed by recognizing that (1) concentration and
temperature obey the same differential equation, (2) the same
mechanical constants (Young's modulus, Poisson ratio) can be used
in both kinds of problem, (3) the lattice dilation coefficient
plays the same role for concentration that is played by the thermal
expansion coefficient with temperature, and (4) the boundary
conditions for a concentration problem can also be taken to be
boundary conditions for a thermal heat flow problem. Thus, one may
set temperature exactly equal to concentration, use Table 1 to see
the correspondence of physical properties, and then calculate
concentration-derived stresses using an existing thermal model.
This allows convenient use of finite element software that is
written for the purpose of thermal stress modeling.
In this way, 3D glass covers having rectangular configurations like
that shown in FIG. 1 have been successfully modeled using the
ANSYS.RTM. finite element thermal modeling software. Through this
modeling, it has been found that when all the surfaces are
ion-exchanged, higher warp is observed along the diagonal axis as
compared to the short and long axes. As a further example, a
rectangular 3D glass cover having a perimeter portion which only
extended upward from the long sides of the rectangle (i.e., a
sled-like structure) was studied. In this case, the expansion of
the edge causes high warp along the ends of long axis.
FIG. 18 illustrates the application of the above techniques to
provide mold contour correction values which achieve improved
dimensional tolerances of 3D glass covers. As shown in this figure,
the method includes the following steps: (1) solve for IOX
diffusion using thermal analogy approach on edges and surfaces; (2)
calculate predicted shape deviation (warp) after IOX diffusion from
given target (CAD) design; (3) inverse the warp values to get
corrected values for the mold's molding surface; and (4) produce a
molding surface having the corrected values. Once the mold is
machined with the corrected values, the warp values after IOX of 3D
glass covers made with the mold are essentially negligible. Note
that as discussed in Example 2 below, in some cases, it may be
desirable not to apply the full IOX correction to the mold surface,
in which case, after ion exchange, the covers will have some
residual IOX warp, but less than they would have had with an
uncorrected mold (see, for example, FIGS. 20 and 21 discussed
below).
Typically, IOX of a 3D glass cover results in dome-shaped warp
wherein when looking at the concave side of the shape, the center
lifts up and edges move down. Accordingly, the inverse values used
to correct the mold will typically result in a mold surface that
has an inverse dome shape, i.e., the portion of the mold that
produces the flat or nearly flat central portion of the 3D glass
cover is not flat. However, the as-molded 3D glass cover with its
inverse dome-shaped contour acquired from the mold becomes flat or
nearly flat after IOX, as is desired. In addition to warp, like
with 2D parts, IOX of 3D glass covers also results in an overall
increase in part size.
The mathematical procedures described above can be readily
implemented using a variety of computer equipment, including
personal computers, workstations, mainframes, etc. Output from the
procedures can be in electronic and/or hard copy form, and can be
displayed in a variety of formats, including in tabular and
graphical form. Software code, including data input routines for
commercial software packages, can be stored and/or distributed in a
variety of forms, e.g., on a hard drive, diskette, CD, flash drive,
etc.
Molding of 3D glass articles in accordance with the present
disclosure can be performed using equipment now known or
subsequently developed. Likewise, for ion exchange treatments, bath
solutions that are now known or subsequently developed can be used.
Along the same lines, the glass articles can have a variety of
compositions suitable for ion diffusion processing now known or
subsequently developed.
Without intending to limit it in any manner, the invention will be
further illustrated by the following examples. The glass used in
these examples was Code 2317 glass, commercially available from
Corning, Inc. (i.e., Corning Gorilla.RTM. glass). This
representative, ion-exchangeable glass was also used in the
simulations of FIGS. 7-15.
EXAMPLE 1
This example illustrates the importance of including the effects of
ion exchange through the edge of a 3D glass cover when predicting
IOX shape changes.
FIG. 19 shows two warp predictions, one (the upper portion of the
figure) that included ion exchange through the edge of the 3D glass
cover and the other (the lower portion of the figure) that did not.
In this figure, the reference number 105 represents the target
shape, while the reference numbers 106 and 107 represent the
predicted shapes with and without edge IOX, respectively.
As can be seen in FIG. 19, failure to include the effects of edge
IOX results in a substantial underestimate of IOX warp, the
magnitude of the predicted/estimated warp without including edge
IOX being .about.14 microns, while with the effects of edge IOX
included, it increases to .about.130 microns. Consequently, mold
compensation based on predicted shape 107 would result in
manufacture of 3D glass covers with substantial deviation from CAD
after IOX that would be unacceptable to customers for such covers,
while mold compensation based on predicted shape 106 would result
in glass covers that would meet customer specifications.
EXAMPLE 2
This example illustrates an application of the process steps of
FIG. 18. In particular, the example compares a predicted shape for
a 3D glass cover made with a corrected (compensated) mold versus
the predicted shape for the same cover made with an uncorrected
mold.
The results are shown in FIGS. 20 and 21, where FIG. 20 shows the
benefits of mold correction for the perimeter portion of the glass
cover (identified by the variable "curve length" in this figure)
and FIG. 21 shows the benefits for the central portion (identified
by the variable "flat length" in this figure). The "without
correction" data assumes the 3D glass cover was built with a mold
whose molding surface matched the target shape, while the "with
correction" data assumes the part was built with a mold whose
molding surface was corrected to take account of IOX warp,
including IOX warp arising from ions passing through the edge of
the part.
It should be noted that the "with correction" data is for a mold
that was not fully corrected for IOX warp. This data thus
illustrates an embodiment of the present disclosure wherein
predicted warp is not completely cancel out when modifying a mold,
but less than the full predicted correction is made to accommodate
other considerations that may need to be taken into account in a
mold design, e.g., the cost of machining complex surfaces and/or
thermal relaxation considerations.
Both the "without correction" and the "with correction"
calculations included the effects of edge IOX, the difference
between the data being the "as-molded" shape at the beginning of
the IOX process, the "without correction" as-molded shape being the
target shape and the "with correction" as-molded shape being the
target shape with the partial correction for IOX warp.
As can be seen in FIGS. 20 and 21, the post-IOX shapes of 3D glass
covers can be substantially improved using the mold correction
techniques disclosed herein. For example, quantitatively, the data
of FIGS. 20 and 21 show a warp reduction after performing IOX from
about 90 microns to about 10 microns.
EXAMPLE 3
This example further illustrates the process steps of FIG. 18. In
particular, the example compares a 3D glass cover made with a
corrected (compensated) mold versus the same cover made with an
uncorrected mold.
FIG. 22 shows the shape change resulting from IOX for a dish shape
part whose dimensions were 110 mm.times.55 mm.times.2 mm. The
measured as-formed shape for the uncorrected mold, i.e., the mold
whose mold surface corresponded to the target shape, is shown by
curve 108 in FIG. 22. As can be seen, the maximum magnitude of the
flatness deviation of the central portion of the as-molded part was
less than 20 microns (the vertical axes in FIGS. 22 and 23 are in
millimeters). After IOX, the part warped strongly as shown by curve
109, the maximum magnitude of the flatness deviation of the central
portion now being greater than 80 microns.
Thereafter, the mold was corrected in accordance with the process
of FIG. 18, i.e., based on the predicted IOX warp for the part, and
again used to make a 3D glass cover. The results of measurements on
the as-molded (as-formed) part and the part after IOX are shown in
FIG. 23, where curve 110 shows the shape of the as-molded part and
curve 111 shows the shape of the post-IOX part. As can be seen, by
correcting the mold, the flatness of the central portion of the
as-molded cover was made worse, i.e., the maximum magnitude of the
flatness deviation was now 90 microns, but the flatness of the
central portion after IOX was now better, i.e., the maximum
magnitude of the flatness deviation was now 40 microns.
Further experiments were performed in which compensated molds were
used to produce 3D glass covers of various configurations. Cases
were studied in which the correction removed some or essentially
all warp at the center of the part after IOX. The second and third
columns of Table 2 set forth the predicted residual IOX warp at the
center of the part ("model predictions") and the measured warp at
that location ("experimental measurements"). As can be seen, the
predicted and measured values correspond closely thus demonstrating
the ability of the mold compensation procedures disclosed herein to
control IOX warp of 3D glass covers.
A variety of modifications that do not depart from the scope and
spirit of the invention will be evident to persons of ordinary
skill in the art from the foregoing disclosure. The following
claims are intended to cover the specific embodiments set forth
herein as well as modifications, variations, and equivalents of
those embodiments.
TABLE-US-00001 TABLE 1 Analogous Property Dimensions thermal value
Young's Modulus E MPa Same Poisson ratio .nu. Same Nominal surface
concentration (C.sub.surf) mole % Temperature Nominal initial
concentration C.sub.base mole % Initial temperature Lattice
dilation constant B 1/(mole %) CTE Diffusivity D m.sup.2/sec
Thermal diffusivity k/(.rho.C.sub.p) Density .rho. and specific
heat C.sub.p NA .rho.C.sub.p = 1
TABLE-US-00002 TABLE 2 Center Warp Values in Microns Glass Model
Experimental Cover Shape Predictions Measurements Circular Dish 13
17 Rectangular Dish 1 92 99 Rectangular Dish 2 53 42 Rectangular
Dish 3 70 60 Rectangular Dish 4* 75 63 *Bend radius on only two
sides.
* * * * *